What is alpha? What happens when it goes to infinity? How come the roots go from the poles of PC to the zeros of PC?

Shreesh P. Mysore, Nov 25, 2002

alpha is simply a parameter, typically the loop gain (but it can be any parameter). Let us call it K for convenience. The aim of the root locus method is to plot the roots of 1+KPC (i.e, solutions to 1+KPC=0) as K varies. Rewrite 1+KPC=0 as PC=-1/K.

When K=0, we see that we need PC to be "infinite" for this equation to be satisfied. This happens at the poles of PC. Thus when K=0, the roots of 1+KPC are at the poles of PC.

When K goes to infinity, we need PC to be zero for the equation to be satisfied. This can happen in two ways -(a) if the roots are at the zeros of PC or (b) the roots approach "asymptotes" defined appropriately (CDS 110). For now, just consider case (a). So when K goes to infinity, we have that the roots go to the zeros of PC.

Thus the statement that "the roots of 1+KPC go from the poles of PC to the zeros of PC" (it is understood in this context that we are increasing K from zero to infinity).